More on the uniqueness of the Lyapunov scaling factors (Q1112919)

A Lyapunov scaling factor (LSF) of a real \(n\times n\) matrix A is a positive diagonal matrix D such that \(AD+DA^ T>0\) (\(\geq 0)\); then A is called Lyapunov diagonally stable (diagonally semistable), \(>(\geq)\) meaning positive definiteness (positive semidefiniteness). In extension of work by \textit{D. Hershkowitz} and \textit{H. Schneider} [Linear Multilinear Algebra 17, 203-226 (1985; Zbl 0593.15016)] the authors give sufficient conditions for a Lyapunov diagonally semistable A to have a unique (up to multiplication by a scalar) LSF. Also a necessary and sufficient condition for the uniqueness of an LSF is given (assuming existence).